AI 中文总结
综述图中多重集可解性参数，涵盖多重集维数等概念，证明直径为二的图等的外部多重集维数下界及局部多重集维数相关新结果，解决文献中两个开放问题，还列出开放问题清单供未来研究。
AI 中文摘要
度量维数在度量图论中是最重要且研究最广泛的主题之一，它有许多变体并在其他领域有众多应用。本文综述了通过考虑来自固定顶点的距离多重集而非原始版本中的向量来实现可解性的结果。讨论的概念包括多重集维数、外部多重集维数、局部多重集维数、边多重集维数和\(k -\)多重集反维数。证明了直径为二的图和无边图与其他图的并图的外部多重集维数的精确下界，解决了文献中的两个开放问题。还证明了局部多重集维数等于二的图的新结果，特别是在块图中对这类图进行了特征刻画。最后，汇编了文献中的开放问题列表，并添加了几个新问题以供未来研究。
英文摘要
The metric dimension, which has lots of variants and numerous applications in other fields, is one of the most important and most extensively studied topics in metric graph theory. Results in which resolvability is achieved by considering multisets of distances from a fixed vertex, instead of vectors as in the original version, are surveyed. The concepts discussed are multiset dimension, outer multiset dimension, local multiset dimension, edge multiset dimension, and $k$-multiset antidimension. Along the way, sharp lower bounds on the outer multiset dimension of diameter two graphs and join graphs with edgeless graphs are proved, which solves two open problems from the literature. New results on graphs with local multiset dimension equal to two are also proved. In particular, such graphs are characterized among block graphs. Finally, a list of open problems from the literature is compiled, and several new problems are added to the list for future research.
Comments31 pages
